Transcript Compare And Order Non-rational numbers - Math GR. 6-8

Compare And Order
Rational Numbers
Benchmark MA.6.A.5.3
Estimate the results of
computations with fractions,
decimals, and percents and judge
the reasonableness of the results
Reasonableness
Defined as:
•having good sense and sound judgment
•being prudent and sensible
•being plausible or acceptable
Reasonableness Example
Reasonableness:
Proper Fractions
A proper fraction is a fraction
that is less than 1 and greater than
zero.
Decimals
A decimal is the representation of
a real number using the base 10
and decimal notation, such as
201.4, 3.89, or 0.0006.
Decimals vs Fractions
A "decimal" is a fraction whose denominator we do
not write but which we understand to be a power
of ten.
For example,
0.4 is read as four tenths or 4/10
0.74 is read as seventy four hundredths or 74/100
Percents
A percent is a way of expressing a
number as a fraction of 100.
Per cent means "per hundred”.
For example,
20% is read as twenty percent or 20/100
3% is read as three percent or 3/100
Fractions, Decimals, Percents
All represent a part of a whole.
•A fraction is based on the number into which the
whole is divided (the denominator). The numerator
(the top) is the PART, the denominator (the bottom)
is the WHOLE.
•A decimal is based on the number in terms of tenths,
hundredths, thousandths, etc.
•A percent is based on the number in terms of 100.
Judging Reasonableness
If you were to take the whole number portion of each fraction:
Can you eliminate any answer
choices? Why?
Add those whole numbers (2 + 4 = 6). She started with 10.
Subtract 10 – 6 = 4. Choices B & D are unreasonable.
Benchmark MA.6.A.5.2
Compare and order fractions,
decimals, and percents, including
finding their approximate
location on a number line.
What are rational numbers?
Rational numbers are parts of a whole. They can be
expressed as a fraction (1/4), a decimal (0.25) or a
percent (25%).
Rational numbers can be plotted on a number line.
Egyptian Fractions
• The ancient Egyptians only used fractions
of the form 1/n .
• All fractions had to be represented as a sum
of such unit fractions.
• This makes it easier to compare fractions.
Egyptian Fractions
How it worked:
Egyptians had a notation for 1/2 and 1/3 and
1/ and so on (these were called reciprocals
4
or unit fractions since they are 1/n for some
number n).
Egyptian Fractions
How did they write 3/4?
They were able to write any fraction as a
sum of unit fractions
3/
4
= 1/2 + 1/4
Egyptian Fractions
So suppose Faith has 5 loaves of bread to
share among the 8 people.
Egyptian Fractions
Faith sees that she can give each person half
a loaf, with one loaf left over.
1
2
7
3
5
4
8
?
6
Egyptian Fractions
Faith takes the one loaf that is left and
divides it into 8 pieces, so each person gets
half a loaf and an eighth of a loaf.
1
5
2
6
3
4
7
8
Egyptian Fractions
Using Egyptian Fractions we see that
5/
8
= 1/2 + 1/8
Egyptian Fractions
Suppose Faith had 3 loaves to share between
4 people. How would she do it?
1
2
3
4
1
3
2
4
each person gets half a loaf and a fourth of
1/ + 1/ = 3/
a loaf.
2
4
4
Egyptian Fractions
What if she had 4 loaves to share between 5
people?
1
2
3
4
5
each person gets half a loaf
?
?
Egyptian Fractions
What if she had 4 loaves to share between 5 people?
There is ½ of a loaf and 1 loaf left.
1 2
3 4 5 ?
each person gets a fourth a loaf in addition
1/ + 1/ + ?
to their a half of a loaf.
2
4
Egyptian Fractions
What if she had 4 loaves to share between 5 people?
There is 1/5 of a loaf left.
1
2
3
4
5
each person gets a fifth of the fourth that was
left in addition to their a half of a loaf and a
1/ + 1/ + 1/ = 16/
quarter of a loaf.
2
4
20
20
Using Egyptian Fractions to
Compare Fractions
Which is larger:
3/ or 4/ ?
4
5
Using Egyptian Fractions to
Compare Fractions
Using Egyptian fractions we write 3/4 as a sum
of unit fractions:
3/ = 2/ + 1/ = 1/ + 1/
4
4
4
2
4
Using Egyptian Fractions to
Compare Fractions
Using Egyptian fractions we write 4/5 as a
sum of unit fractions:
4/ = 1/ + 3/
5
2
10
= 1/2 + 6/20
= 1/2 + 4/20 + 1/20
= 1/2 + 1/4 + 1/20
Using Egyptian Fractions to
Compare Fractions
3/
4
4/
5
4/
5
= 1/2 + 1/4
= 1/2 + 1/4 + 1/20
is the larger than 3/4 by exactly 1/20
Comparing Fractions using Decimals
Convert the fractions to decimals:
3/ =75/
4
100 or 0.75
4/ = 80/
5
100 or 0.80
80 (hundredths) is bigger than 75
(hundredths) therefore 4/5 is bigger than 3/4
Ordering Rational Numbers
One way to order rational numbers is
graphing them on a number line.
On a number line, the rational number to the
right of another rational number is greater.
least
greatest
Ordering Rational Numbers
A second method is to convert all rational numbers
to decimals. Place the following numbers in order
largest to smallest: 1.112, 0.234, 1.056, 0.45
1
0
1
0
.
.
.
.
1
2
0
4
1
3
5
5
2
4
6
0
Place zero in
empty spots
Ordering Decimal Numbers
Place the following numbers in order largest to
smallest: 1.112, 0.234, 1.056, 0.45
1
0
1
0
.
.
.
.
1
2
0
4
1
3
5
5
2
4
6
0
largest
smallest
Ordering Decimal Numbers
Place the following numbers in order largest to
smallest: 1.112, 0.234, 1.056, 0.45
1
1
0
0
.
.
.
.
1
0
4
2
1
5
5
3
2
6
0
4
largest
smallest
Ordering Rational Numbers
Besides using number line, and decimals, you
can use the common denominator method.
Convert all rational numbers to fractions with
common denominators. Place the following
numbers in order from smallest to largest:
2/ , 11/ , 3/ 15/
5
2
4,
6
Ordering Rational Numbers
Place the following numbers in order from
smallest to largest:
2/ , 11/ , 3/ 15/
5
2
4,
6
2/ = 24/
5
60
11/2 = 3/2 = 90/60
3/ = 45/
4
60
15/6 = 11/6 = 110/60
smallest
largest
Ordering Rational Numbers
The order from smallest to largest is
2/
5
,
3/ 11/
4,
2
,
smallest
2/ = 24/
5
60
11/2 = 3/2 = 90/60
3/ = 45/
4
60
15/6 = 11/6 = 110/60
largest
5
1 /6
Guided Practice #1
Which is the greater number, 23% or 2.5
Percent means per hundred. So, 23% is the
same as 23/100 or 0 .23
0 .23 is smaller than 2.5
2.5 > 23%
Guided Practice #2
Graph this set of numbers on a number line.
What is the order of the set of numbers from
least to greatest?
-1.5, 2, 21/2 , -2 5/6 , 1.4, 25%
Guided Practice #2
Step 1: Draw a number line from -3 to 3 with
equal intervals.
-25/6
-1.5
25%
1.4
2
21/2
Step 2: Plot each point asked for in the
problem: -1.5, 2, 21/2 , -2 5/6 , 1.4, 25%
Guided Practice #2
Step 3: Use the points plotted on the number
line to write the numbers in order from least
to greatest.
-25/6
-1.5
25%
1.4
-25/6 , -1.5, 25%, 1.4, 2, 21/2
2
21/2
Practice Activity #1
Silence is Golden
Practice Activity #2
Hexagon Domino Puzzle